Physics FormularyPhysics Formulary by ir. J.C.A. Wevers III 7.6 Interaction between molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 8 Thermodynamics

Physics Formulary

By ir. J.C.A. Wevers

c© 1995, 2005 J.C.A. Wevers Version: April 14, 2005

Dear reader,

This document contains a 108 page LATEX file which contains a lot equations in physics. It is written at advancedundergraduate/postgraduate level. It is intended to be a short reference for anyone who works with physics andoften needs to look up equations.

This, and a Dutch version of this file, can be obtained from the author, Johan Wevers([email protected] ).

It can also be obtained on the WWW. Seehttp://www.xs4all.nl/˜johanw/index.html , wherealso a Postscript version is available.

If you find any errors or have any comments, please let me know. I am always open for suggestions andpossible corrections to the physics formulary.

This document is Copyright 1995, 1998 by J.C.A. Wevers. All rights are reserved. Permission to use, copyand distribute this unmodified document by any means and for any purposeexcept profit purposesis herebygranted. Reproducing this document by any means, included, but not limited to, printing, copying existingprints, publishing by electronic or other means, implies full agreement to the above non-profit-use clause,unless upon explicit prior written permission of the author.

This document is provided by the author “as is”, with all its faults. Any express or implied warranties, in-cluding, but not limited to, any implied warranties of merchantability, accuracy, or fitness for any particularpurpose, are disclaimed. If you use the information in this document, in any way, you do so at your own risk.

The Physics Formulary is made with teTEX. This pdf file is made with pdfTEX.

Johan [email protected]

http://www.xs4all.nl/~johanwmailto:[email protected]

Contents

Contents I

Physical Constants 1

1 Mechanics 21.1 Point-kinetics in a fixed coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Relative motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Point-dynamics in a fixed coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Force, (angular)momentum and energy . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Conservative force fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.3 Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.4 Orbital equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.5 The virial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Point dynamics in a moving coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.1 Apparent forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.2 Tensor notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Dynamics of masspoint collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5.1 The centre of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5.2 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.6 Dynamics of rigid bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6.1 Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6.2 Principal axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6.3 Time dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.7 Variational Calculus, Hamilton and Lagrange mechanics . . . . . . . . . . . . . . . . . . . . 61.7.1 Variational Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7.2 Hamilton mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7.3 Motion around an equilibrium, linearization . . . . . . . . . . . . . . . . . . . . . . . 71.7.4 Phase space, Liouville’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7.5 Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Electricity & Magnetism 92.1 The Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Force and potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Energy of the electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5.1 Electromagnetic waves in vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5.2 Electromagnetic waves in matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.6 Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 Electric currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.8 Depolarizing field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.9 Mixtures of materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

I

II Physics Formulary by ir. J.C.A. Wevers

3 Relativity 133.1 Special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 The Lorentz transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.2 Red and blue shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.3 The stress-energy tensor and the field tensor . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 General relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.1 Riemannian geometry, the Einstein tensor . . . . . . . . . . . . . . . . . . . . . . . . 143.2.2 The line element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.3 Planetary orbits and the perihelion shift . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.4 The trajectory of a photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.5 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.6 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Oscillations 184.1 Harmonic oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Mechanic oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 Electric oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4 Waves in long conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.5 Coupled conductors and transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.6 Pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Waves 205.1 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2 Solutions of the wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2.1 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2.2 Spherical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2.3 Cylindrical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2.4 The general solution in one dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.3 The stationary phase method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.4 Green functions for the initial-value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.5 Waveguides and resonating cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.6 Non-linear wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Optics 246.1 The bending of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2 Paraxial geometrical optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.2.1 Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2.2 Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2.3 Principal planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2.4 Magnification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.3 Matrix methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.4 Aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.5 Reflection and transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.6 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.7 Prisms and dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.8 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.9 Special optical effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.10 The Fabry-Perot interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7 Statistical physics 307.1 Degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2 The energy distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.3 Pressure on a wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.4 The equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.5 Collisions between molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Physics Formulary by ir. J.C.A. Wevers III

7.6 Interaction between molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8 Thermodynamics 338.1 Mathematical introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.3 Thermal heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.4 The laws of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.5 State functions and Maxwell relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.6 Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.7 Maximal work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.8 Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.9 Thermodynamic potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.10 Ideal mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.11 Conditions for equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.12 Statistical basis for thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.13 Application to other systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

9 Transport phenomena 399.1 Mathematical introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.2 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.3 Bernoulli’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.4 Characterising of flows by dimensionless numbers . . . . . . . . . . . . . . . . . . . . . . . . 419.5 Tube flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.6 Potential theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.7 Boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9.7.1 Flow boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.7.2 Temperature boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9.8 Heat conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.9 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.10 Self organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

10 Quantum physics 4510.1 Introduction to quantum physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10.1.1 Black body radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.1.2 The Compton effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.1.3 Electron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10.2 Wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.3 Operators in quantum physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.4 The uncertainty principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.5 The Schr̈odinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.6 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.7 The tunnel effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.8 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.9 Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.10 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.11 The Dirac formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.12 Atomic physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

10.12.1 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.12.2 Eigenvalue equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.12.3 Spin-orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.12.4 Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10.13 Interaction with electromagnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.14 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10.14.1 Time-independent perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . 5010.14.2 Time-dependent perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . 51

IV Physics Formulary by ir. J.C.A. Wevers

10.15 N-particle systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.15.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.15.2 Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

10.16 Quantum statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

11 Plasma physics 5411.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.2 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.3 Elastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

11.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5511.3.2 The Coulomb interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.3.3 The induced dipole interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.3.4 The centre of mass system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.3.5 Scattering of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

11.4 Thermodynamic equilibrium and reversibility . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.5 Inelastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

11.5.1 Types of collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.5.2 Cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

11.6 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.7 The Boltzmann transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5911.8 Collision-radiative models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6011.9 Waves in plasma’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

12 Solid state physics 6212.1 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.2 Crystal binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.3 Crystal vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

12.3.1 A lattice with one type of atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.3.2 A lattice with two types of atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.3.3 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.3.4 Thermal heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

12.4 Magnetic field in the solid state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6512.4.1 Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6512.4.2 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6512.4.3 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

12.5 Free electron Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.5.1 Thermal heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.5.2 Electric conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.5.3 The Hall-effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6712.5.4 Thermal heat conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

12.6 Energy bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6712.7 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6712.8 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

12.8.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6812.8.2 The Josephson effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6912.8.3 Flux quantisation in a superconducting ring . . . . . . . . . . . . . . . . . . . . . . . 6912.8.4 Macroscopic quantum interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7012.8.5 The London equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7012.8.6 The BCS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Physics Formulary by ir. J.C.A. Wevers V

13 Theory of groups 7113.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

13.1.1 Definition of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7113.1.2 The Cayley table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7113.1.3 Conjugated elements, subgroups and classes . . . . . . . . . . . . . . . . . . . . . . . 7113.1.4 Isomorfism and homomorfism; representations . . . . . . . . . . . . . . . . . . . . . 7213.1.5 Reducible and irreducible representations . . . . . . . . . . . . . . . . . . . . . . . . 72

13.2 The fundamental orthogonality theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.2.1 Schur’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.2.2 The fundamental orthogonality theorem . . . . . . . . . . . . . . . . . . . . . . . . . 7213.2.3 Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

13.3 The relation with quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.3.1 Representations, energy levels and degeneracy . . . . . . . . . . . . . . . . . . . . . 7313.3.2 Breaking of degeneracy by a perturbation . . . . . . . . . . . . . . . . . . . . . . . . 7313.3.3 The construction of a base function . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.3.4 The direct product of representations . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.3.5 Clebsch-Gordan coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.3.6 Symmetric transformations of operators, irreducible tensor operators . . . . . . . . . . 7413.3.7 The Wigner-Eckart theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

13.4 Continuous groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.4.1 The 3-dimensional translation group . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.4.2 The 3-dimensional rotation group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.4.3 Properties of continuous groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

13.5 The group SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7713.6 Applications to quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

13.6.1 Vectormodel for the addition of angular momentum . . . . . . . . . . . . . . . . . . . 7713.6.2 Irreducible tensor operators, matrixelements and selection rules . . . . . . . . . . . . 78

13.7 Applications to particle physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

14 Nuclear physics 8114.1 Nuclear forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8114.2 The shape of the nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8214.3 Radioactive decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8214.4 Scattering and nuclear reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

14.4.1 Kinetic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8314.4.2 Quantum mechanical model for n-p scattering . . . . . . . . . . . . . . . . . . . . . . 8314.4.3 Conservation of energy and momentum in nuclear reactions . . . . . . . . . . . . . . 84

14.5 Radiation dosimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

15 Quantum field theory & Particle physics 8515.1 Creation and annihilation operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8515.2 Classical and quantum fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8515.3 The interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8615.4 Real scalar field in the interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8615.5 Charged spin-0 particles, conservation of charge . . . . . . . . . . . . . . . . . . . . . . . . 8715.6 Field functions for spin-12 particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8715.7 Quantization of spin-12 fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8815.8 Quantization of the electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8915.9 Interacting fields and the S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8915.10 Divergences and renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9015.11 Classification of elementary particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9015.12 P and CP-violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9215.13 The standard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

15.13.1 The electroweak theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9315.13.2 Spontaneous symmetry breaking: the Higgs mechanism . . . . . . . . . . . . . . . . 94

VI Physics Formulary by ir. J.C.A. Wevers

15.13.3 Quantumchromodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9415.14 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9515.15 Unification and quantum gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

16 Astrophysics 9616.1 Determination of distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9616.2 Brightness and magnitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9616.3 Radiation and stellar atmospheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9716.4 Composition and evolution of stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9716.5 Energy production in stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

The∇-operator 99

The SI units 100

Physical Constants

Name Symbol Value UnitNumberπ π 3.14159265358979323846Number e e 2.71828182845904523536

Euler’s constant γ = limn→∞

(n∑

k=1

1/k − ln(n))

= 0.5772156649

Elementary charge e 1.60217733 · 10−19 CGravitational constant G, κ 6.67259 · 10−11 m3kg−1s−2Fine-structure constant α = e2/2hcε0 ≈ 1/137Speed of light in vacuum c 2.99792458 · 108 m/s (def)Permittivity of the vacuum ε0 8.854187 · 10−12 F/mPermeability of the vacuum µ0 4π · 10−7 H/m(4πε0)−1 8.9876 · 109 Nm2C−2

Planck’s constant h 6.6260755 · 10−34 JsDirac’s constant h̄ = h/2π 1.0545727 · 10−34 JsBohr magneton µB = eh̄/2me 9.2741 · 10−24 Am2Bohr radius a0 0.52918 ÅRydberg’s constant Ry 13.595 eVElectron Compton wavelengthλCe = h/mec 2.2463 · 10−12 mProton Compton wavelength λCp = h/mpc 1.3214 · 10−15 mReduced mass of the H-atom µH 9.1045755 · 10−31 kgStefan-Boltzmann’s constant σ 5.67032 · 10−8 Wm−2K−4Wien’s constant kW 2.8978 · 10−3 mKMolar gasconstant R 8.31441 J·mol−1·K−1Avogadro’s constant NA 6.0221367 · 1023 mol−1Boltzmann’s constant k = R/NA 1.380658 · 10−23 J/KElectron mass me 9.1093897 · 10−31 kgProton mass mp 1.6726231 · 10−27 kgNeutron mass mn 1.674954 · 10−27 kgElementary mass unit mu = 112m(

126 C) 1.6605656 · 10−27 kg

Nuclear magneton µN 5.0508 · 10−27 J/TDiameter of the Sun D� 1392 · 106 mMass of the Sun M� 1.989 · 1030 kgRotational period of the Sun T� 25.38 daysRadius of Earth RA 6.378 · 106 mMass of Earth MA 5.976 · 1024 kgRotational period of Earth TA 23.96 hoursEarth orbital period Tropical year 365.24219879 daysAstronomical unit AU 1.4959787066 · 1011 mLight year lj 9.4605 · 1015 mParsec pc 3.0857 · 1016 mHubble constant H ≈ (75± 25) km·s−1·Mpc−1

1

Chapter 1

Mechanics

1.1 Point-kinetics in a fixed coordinate system

1.1.1 Definitions

The position~r, the velocity~v and the acceleration~a are defined by:~r = (x, y, z), ~v = (ẋ, ẏ, ż), ~a = (ẍ, ÿ, z̈).The following holds:

s(t) = s0 +∫|~v(t)|dt ; ~r(t) = ~r0 +

∫~v(t)dt ; ~v(t) = ~v0 +

∫~a(t)dt

When the acceleration is constant this gives:v(t) = v0 + at ands(t) = s0 + v0t+ 12at2.

For the unit vectors in a direction⊥ to the orbit~et and parallel to it~en holds:

~et =~v

|~v|=d~r

ds~̇et =

v

ρ~en ; ~en =

~̇et

|~̇et|

For thecurvaturek and theradius of curvatureρ holds:

~k =d~etds

=d2~r

ds2=∣∣∣∣dϕds

∣∣∣∣ ; ρ = 1|k|1.1.2 Polar coordinates

Polar coordinates are defined by:x = r cos(θ), y = r sin(θ). So, for the unit coordinate vectors holds:~̇er = θ̇~eθ, ~̇eθ = −θ̇~er

The velocity and the acceleration are derived from:~r = r~er, ~v = ṙ~er +rθ̇~eθ,~a = (r̈−rθ̇2)~er +(2ṙθ̇+rθ̈)~eθ.

1.2 Relative motion

For the motion of a point D w.r.t. a point Q holds:~rD = ~rQ +~ω × ~vQω2

with ~QD = ~rD − ~rQ andω = θ̇.

Further holds:α = θ̈. ′ means that the quantity is defined in a moving system of coordinates. In a movingsystem holds:~v = ~vQ + ~v ′ + ~ω × ~r ′ and~a = ~aQ + ~a ′ + ~α× ~r ′ + 2~ω × ~v ′ + ~ω × (~ω × ~r ′)with ~ω × (~ω × ~r ′) = −ω2~r ′n

1.3 Point-dynamics in a fixed coordinate system

1.3.1 Force, (angular)momentum and energy

Newton’s 2nd law connects the force on an object and the resulting acceleration of the object where themo-mentumis given by~p = m~v:

~F (~r,~v, t) =d~p

dt=d(m~v )dt

= md~v

dt+ ~v

dm

dt

m=const= m~a

2

Chapter 1: Mechanics 3

Newton’s 3rd law is given by:~Faction = −~Freaction.For the powerP holds:P = Ẇ = ~F ·~v. For the total energyW , the kinetic energyT and the potential energyU holds:W = T + U ; Ṫ = −U̇ with T = 12mv

2.

Thekick ~S is given by:~S = ∆~p =∫

~Fdt

The workA, delivered by a force, isA =

2∫1

~F · d~s =2∫

1

F cos(α)ds

The torque~τ is related to the angular momentum~L: ~τ = ~̇L = ~r × ~F ; and~L = ~r × ~p = m~v × ~r, |~L| = mr2ω. The following equation is valid:

τ = −∂U∂θ

Hence, the conditions for a mechanical equilibrium are:∑ ~Fi = 0 and∑~τi = 0.

The force of frictionis usually proportional to the force perpendicular to the surface, except when the motionstarts, when a threshold has to be overcome:Ffric = f · Fnorm · ~et.

1.3.2 Conservative force fields

A conservative force can be written as the gradient of a potential:~Fcons = −~∇U . From this follows that∇× ~F = ~0. For such a force field also holds:∮

~F · d~s = 0 ⇒ U = U0 −r1∫

r0

~F · d~s

So the work delivered by a conservative force field depends not on the trajectory covered but only on thestarting and ending points of the motion.

1.3.3 Gravitation

The Newtonian law of gravitation is (in GRT one also usesκ instead ofG):

~Fg = −Gm1m2r2

~er

The gravitational potential is then given byV = −Gm/r. From Gauss law it then follows:∇2V = 4πG%.

1.3.4 Orbital equations

If V = V (r) one can derive from the equations of Lagrange forφ the conservation of angular momentum:

∂L∂φ

=∂V

∂φ= 0⇒ d

dt(mr2φ) = 0⇒ Lz = mr2φ = constant

For the radial position as a function of time can be found that:(dr

dt

)2=

2(W − V )m

− L2

m2r2

The angular equation is then:

φ− φ0 =r∫

0

[mr2

L

√2(W − V )

m− L

2

m2r2

]−1dr

r−2field= arccos

(1 +

1r −

1r0

1r0

+ km/L2z

)

If F = F (r): L =constant, ifF is conservative:W =constant, if~F ⊥ ~v then∆T = 0 andU = 0.

4 Physics Formulary by ir. J.C.A. Wevers

Kepler’s orbital equations

In a force fieldF = kr−2, the orbits are conic sections with the origin of the force in one of the foci (Kepler’s1st law). The equation of the orbit is:

r(θ) =`

1 + ε cos(θ − θ0), or: x2 + y2 = (`− εx)2

with

` =L2

Gµ2Mtot; ε2 = 1 +

2WL2

G2µ3M2tot= 1− `

a; a =

`

1− ε2=

k

2Wa is half the length of the long axis of the elliptical orbit in case the orbit is closed. Half the length of the shortaxis isb =

√a`. ε is theexcentricityof the orbit. Orbits with an equalε are of equal shape. Now, 5 types of

orbits are possible:

1. k < 0 andε = 0: a circle.

2. k < 0 and0 < ε < 1: an ellipse.

3. k < 0 andε = 1: a parabole.

4. k < 0 andε > 1: a hyperbole, curved towards the centre of force.

5. k > 0 andε > 1: a hyperbole, curved away from the centre of force.

Other combinations are not possible: the total energy in a repulsive force field is always positive soε > 1.

If the surface between the orbit covered betweent1 andt2 and the focus C around which the planet moves isA(t1, t2), Kepler’s 2nd law is

A(t1, t2) =LC2m

(t2 − t1)

Kepler’s 3rd law is, withT the period andMtot the total mass of the system:

T 2

a3=

4π2

GMtot

1.3.5 The virial theorem

The virial theorem for one particle is:

〈m~v · ~r〉 = 0⇒ 〈T 〉 = − 12〈~F · ~r

〉= 12

〈rdU

dr

〉= 12n 〈U〉 if U = −

k

rn

The virial theorem for a collection of particles is:

〈T 〉 = − 12

〈 ∑particles

~Fi · ~ri +∑pairs

~Fij · ~rij

〉These propositions can also be written as:2Ekin + Epot = 0.

1.4 Point dynamics in a moving coordinate system

1.4.1 Apparent forces

The total force in a moving coordinate system can be found by subtracting the apparent forces from the forcesworking in the reference frame:~F ′ = ~F − ~Fapp. The different apparent forces are given by:

1. Transformation of the origin:For = −m~aa2. Rotation:~Fα = −m~α× ~r ′

3. Coriolis force:Fcor = −2m~ω × ~v

4. Centrifugal force:~Fcf = mω2~rn ′ = −~Fcp ; ~Fcp = −mv2

r~er


1.4.2 Tensor notation

Transformation of the Newtonian equations of motion toxα = xα(x) gives:

dxα

dt=∂xα

∂x̄βdx̄β

dt;

The chain rule gives:

d

dt

dxα

dt=d2xα

dt2=

d

dt

(∂xα

∂x̄βdx̄β

dt

)=∂xα

∂x̄βd2x̄β

dt2+dx̄β

dt

d

dt

(∂xα

∂x̄β

)so:

d

dt

∂xα

∂x̄β=

∂

∂x̄γ∂xα

∂x̄βdx̄γ

dt=

∂2xα

∂x̄β∂x̄γdx̄γ

dt

This leads to:d2xα

dt2=∂xα

∂x̄βd2x̄β

dt2+

∂2xα

∂x̄β∂x̄γdx̄γ

dt

(dx̄β

dt

)Hence the Newtonian equation of motion

md2xα

dt2= Fα

will be transformed into:

m

{d2xα

dt2+ Γαβγ

dxβ

dt

dxγ

dt

}= Fα

The apparent forces are taken from he origin to the effect side in the wayΓαβγdxβ

dt

dxγ

dt.

1.5 Dynamics of masspoint collections

1.5.1 The centre of mass

The velocity w.r.t. the centre of mass~R is given by~v− ~̇R. The coordinates of the centre of mass are given by:

~rm =∑mi~ri∑mi

In a 2-particle system, the coordinates of the centre of mass are given by:

~R =m1~r1 +m2~r2m1 +m2

With ~r = ~r1 − ~r2, the kinetic energy becomes:T = 12MtotṘ2 + 12µṙ

2, with the reduced massµ given by:1µ

=1m1

+1m2

The motion within and outside the centre of mass can be separated:

~̇Loutside = ~τoutside ; ~̇Linside = ~τinside

~p = m~vm ; ~Fext = m~am ; ~F12 = µ~u

1.5.2 Collisions

With collisions, where B are the coordinates of the collision and C an arbitrary other position, holds:~p = m~vmis constant, andT = 12m~v

2m is constant. The changes in therelative velocitiescan be derived from:~S = ∆~p =

µ(~vaft − ~vbefore). Further holds∆~LC = ~CB× ~S, ~p ‖ ~S =constant and~L w.r.t. B is constant.


1.6 Dynamics of rigid bodies

1.6.1 Moment of Inertia

The angular momentum in a moving coordinate system is given by:

~L′ = I~ω + ~L′n

whereI is themoment of inertiawith respect to a central axis, which is given by:

I =∑

i

mi~ri2 ; T ′ = Wrot = 12ωIij~ei~ej =

12Iω

2

or, in the continuous case:

I =m

V

∫r′

2ndV =

∫r′

2ndm

Further holds:Li = Iijωj ; Iii = Ii ; Iij = Iji = −

∑k

mkx′ix′j

Steiner’s theorem is:Iw.r.t.D = Iw.r.t.C +m(DM)2 if axis C‖ axis D.

Object I Object I

Cavern cylinder I = mR2 Massive cylinder I = 12mR2

Disc, axis in plane disc through m I = 14mR2 Halter I = 12µR

2

Cavern sphere I = 23mR2 Massive sphere I = 25mR

2

Bar, axis⊥ through c.o.m. I = 112ml2 Bar, axis⊥ through end I = 13ml

2

Rectangle, axis⊥ plane thr. c.o.m. I = 112m(a2 + b2) Rectangle, axis‖ b thr. m I = ma2

1.6.2 Principal axes

Each rigid body has (at least) 3 principal axes which stand⊥ to each other. For a principal axis holds:

∂I

∂ωx=

∂I

∂ωy=

∂I

∂ωz= 0 so L′n = 0

The following holds:ω̇k = −aijkωiωj with aijk =Ii − IjIk

if I1 ≤ I2 ≤ I3.

1.6.3 Time dependence

For torque of force~τ holds:

~τ ′ = Iθ̈ ;d′′~L′

dt= ~τ ′ − ~ω × ~L′

Thetorque~T is defined by:~T = ~F × ~d.

1.7 Variational Calculus, Hamilton and Lagrange mechanics

1.7.1 Variational Calculus

Starting with:

δ

b∫a

L(q, q̇, t)dt = 0 with δ(a) = δ(b) = 0 and δ(du

dx

)=

d

dx(δu)


the equations of Lagrange can be derived:d

dt

∂L∂q̇i

=∂L∂qi

When there are additional conditions applying to the variational problemδJ(u) = 0 of the typeK(u) =constant, the new problem becomes:δJ(u)− λδK(u) = 0.

1.7.2 Hamilton mechanics

TheLagrangianis given by:L =∑T (q̇i) − V (qi). TheHamiltonian is given by:H =

∑q̇ipi − L. In 2

dimensions holds:L = T − U = 12m(ṙ2 + r2φ̇2)− U(r, φ).

If the used coordinates arecanonicalthe Hamilton equations are the equations of motion for the system:

dqidt

=∂H

∂pi;

dpidt

= −∂H∂qi

Coordinates are canonical if the following holds:{qi, qj} = 0, {pi, pj} = 0, {qi, pj} = δij where{, } is thePoisson bracket:

{A,B} =∑

i

[∂A

∂qi

∂B

∂pi− ∂A∂pi

∂B

∂qi

]The Hamiltonian of a Harmonic oscillator is given byH(x, p) = p2/2m + 12mω

2x2. With new coordinates(θ, I), obtained by the canonical transformationx =

√2I/mω cos(θ) andp = −

√2Imω sin(θ), with inverse

θ = arctan(−p/mωx) andI = p2/2mω + 12mωx2 it follows: H(θ, I) = ωI.

The Hamiltonian of a charged particle with chargeq in an external electromagnetic field is given by:

H =1

2m

(~p− q ~A

)2+ qV

This Hamiltonian can be derived from the Hamiltonian of a free particleH = p2/2m with the transformations~p → ~p − q ~A andH → H − qV . This is elegant from a relativistic point of view: this is equivalent to thetransformation of the momentum 4-vectorpα → pα − qAα. A gauge transformation on the potentialsAαcorresponds with a canonical transformation, which make the Hamilton equations the equations of motion forthe system.

1.7.3 Motion around an equilibrium, linearization

For natural systems around equilibrium the following equations are valid:(∂V

∂qi

)0

= 0 ; V (q) = V (0) + Vikqiqk with Vik =(

∂2V

∂qi∂qk

)0

With T = 12 (Mik q̇iq̇k) one receives the set of equationsMq̈ + V q = 0. If qi(t) = ai exp(iωt) is substituted,this set of equations has solutions ifdet(V − ω2M) = 0. This leads to the eigenfrequencies of the problem:

ω2k =aTk V akaTkMak

. If the equilibrium is stable holds:∀k thatω2k > 0. The general solution is a superposition if

eigenvibrations.

1.7.4 Phase space, Liouville’s equation

In phase space holds:

∇ =

(∑i

∂

∂qi,∑

i

∂

∂pi

)so ∇ · ~v =

∑i

(∂

∂qi

∂H

∂pi− ∂∂pi

∂H

∂qi

)If the equation of continuity,∂t%+∇ · (%~v ) = 0 holds, this can be written as:

{%,H}+ ∂%∂t

= 0


For an arbitrary quantityA holds:dA

dt= {A,H}+ ∂A

∂t

Liouville’s theorem can than be written as:

d%

dt= 0 ; or:

∫pdq = constant

1.7.5 Generating functions

Starting with the coordinate transformation:{Qi = Qi(qi, pi, t)Pi = Pi(qi, pi, t)

one can derive the following Hamilton equations with the new HamiltonianK:

dQidt

=∂K

∂Pi;

dPidt

= − ∂K∂Qi

Now, a distinction between 4 cases can be made:

1. If piq̇i −H = PiQi −K(Pi, Qi, t)−dF1(qi, Qi, t)

dt, the coordinates follow from:

pi =∂F1∂qi

; Pi = −∂F1∂Qi

; K = H +∂F1∂t

2. If piq̇i −H = −ṖiQi −K(Pi, Qi, t) +dF2(qi, Pi, t)


pi =∂F2∂qi

; Qi =∂F2∂Pi

; K = H +∂F2∂t

3. If −ṗiqi −H = PiQ̇i −K(Pi, Qi, t) +dF3(pi, Qi, t)


qi = −∂F3∂pi

; Pi = −∂F3∂Qi

; K = H +∂F3∂t

4. If −ṗiqi −H = −PiQi −K(Pi, Qi, t) +dF4(pi, Pi, t)


qi = −∂F4∂pi

; Qi =∂F4∂pi

; K = H +∂F4∂t

The functionsF1, F2, F3 andF4 are calledgenerating functions.

Chapter 2

Electricity & Magnetism

2.1 The Maxwell equations

The classical electromagnetic field can be described by theMaxwell equations. Those can be written both asdifferential and integral equations:∫∫

© ( ~D · ~n )d2A = Qfree,included ∇ · ~D = ρfree∫∫© ( ~B · ~n )d2A = 0 ∇ · ~B = 0∮~E · d~s = −dΦ

dt∇× ~E = −∂

~B

∂t∮~H · d~s = Ifree,included +

dΨdt

∇× ~H = ~Jfree +∂ ~D

∂t

For the fluxes holds:Ψ =∫∫

( ~D · ~n )d2A, Φ =∫∫

( ~B · ~n )d2A.

The electric displacement~D, polarization~P and electric field strength~E depend on each other according to:

~D = ε0 ~E + ~P = ε0εr ~E, ~P =∑~p0/Vol, εr = 1 + χe, with χe =

np203ε0kT

The magnetic field strength~H, the magnetization~M and the magnetic flux density~B depend on each otheraccording to:

~B = µ0( ~H + ~M) = µ0µr ~H, ~M =∑

~m/Vol, µr = 1 + χm, with χm =µ0nm

20

3kT

2.2 Force and potential

The force and the electric field between 2 point charges are given by:

~F12 =Q1Q2

4πε0εrr2~er ; ~E =

~F

Q

The Lorentzforce is the force which is felt by a charged particle that moves through a magnetic field. Theorigin of this force is a relativistic transformation of the Coulomb force:~FL = Q(~v × ~B ) = l(~I × ~B ).

The magnetic field in pointP which results from an electric current is given by thelaw of Biot-Savart, alsoknown als the law of Laplace. In here,d~l ‖ ~I and~r points fromd~l to P :

d ~BP =µ0I

4πr2d~l × ~er

If the current is time-dependent one has to takeretardationinto account: the substitutionI(t) → I(t − r/c)has to be applied.

The potentials are given by:V12 = −2∫

1

~E · d~s and ~A = 12 ~B × ~r.

9


Here, the freedom remains to apply agauge transformation. The fields can be derived from the potentials asfollows:

~E = −∇V − ∂~A

∂t, ~B = ∇× ~A

Further holds the relation:c2 ~B = ~v × ~E.

2.3 Gauge transformations

The potentials of the electromagnetic fields transform as follows when a gauge transformation is applied:~A′ = ~A−∇f

V ′ = V +∂f

∂t

so the fields~E and ~B do not change. This results in a canonical transformation of the Hamiltonian. Further,the freedom remains to apply a limiting condition. Two common choices are:

1. Lorentz-gauge:∇· ~A+ 1c2∂V

∂t= 0. This separates the differential equations for~A andV : V = − ρ

ε0,

~A = −µ0 ~J .

2. Coulomb gauge:∇ · ~A = 0. If ρ = 0 and ~J = 0 holdsV = 0 and follows ~A from ~A = 0.

2.4 Energy of the electromagnetic field

The energy density of the electromagnetic field is:

dW

dVol= w =

∫HdB +

∫EdD

The energy density can be expressed in the potentials and currents as follows:

wmag = 12

∫~J · ~Ad3x , wel = 12

∫ρV d3x

2.5 Electromagnetic waves

2.5.1 Electromagnetic waves in vacuum

The wave equation Ψ(~r, t) = −f(~r, t) has the general solution, withc = (ε0µ0)−1/2:

Ψ(~r, t) =∫f(~r, t− |~r − ~r ′|/c)

4π|~r − ~r ′|d3r′

If this is written as:~J(~r, t) = ~J(~r ) exp(−iωt) and ~A(~r, t) = ~A(~r ) exp(−iωt) with:

~A(~r ) =µ

4π

∫~J(~r ′)

exp(ik|~r − ~r ′|)|~r − ~r ′|

d3~r ′ , V (~r ) =1

4πε

∫ρ(~r ′)

exp(ik|~r − ~r ′|)|~r − ~r ′|

d3~r ′

A derivation via multipole expansion will show that for the radiated energy holds, ifd, λ� r:

dP

dΩ=

k2

32π2ε0c

∣∣∣∣∫ J⊥(~r ′)ei~k·~rd3r′∣∣∣∣2The energy density of the electromagnetic wave of a vibrating dipole at a large distance is:

w = ε0E2 =p20 sin

2(θ)ω4

16π2ε0r2c4sin2(kr − ωt) , 〈w〉t =

p20 sin2(θ)ω4

32π2ε0r2c4, P =

ck4|~p |2

12πε0

The radiated energy can be derived from thePoynting vector~S: ~S = ~E × ~H = cW~ev. The irradiance is thetime-averaged of the Poynting vector:I = 〈|~S |〉t. The radiation pressureps is given byps = (1 + R)|~S |/c,whereR is the coefficient of reflection.

Chapter 2: Electricity & Magnetism 11

2.5.2 Electromagnetic waves in matter

The wave equations in matter, withcmat = (εµ)−1/2 the lightspeed in matter, are:(∇2 − εµ ∂

2

∂t2− µρ

∂

∂t

)~E = 0 ,

(∇2 − εµ ∂

2

∂t2− µρ

∂

∂t

)~B = 0

give, after substitution of monochromatic plane waves:~E = E exp(i(~k ·~r−ωt)) and ~B = B exp(i(~k ·~r−ωt))the dispersion relation:

k2 = εµω2 +iµω

ρ

The first term arises from the displacement current, the second from the conductance current. Ifk is written inthe formk := k′ + ik′′ it follows that:

k′ = ω√

12εµ

√√√√1 +√1 + 1(ρεω)2

and k′′ = ω√

12εµ

√√√√−1 +√1 + 1(ρεω)2

This results in a damped wave:~E = E exp(−k′′~n ·~r ) exp(i(k′~n ·~r−ωt)). If the material is a good conductor,

the wave vanishes after approximately one wavelength,k = (1 + i)√µω

2ρ.

2.6 Multipoles

Because1

|~r − ~r ′|=

1r

∞∑0

(r′

r

)lPl(cos θ) the potential can be written as:V =

Q

4πε

∑n

knrn

For the lowest-order terms this results in:

• Monopole:l = 0, k0 =∫ρdV

• Dipole: l = 1, k1 =∫r cos(θ)ρdV

• Quadrupole:l = 2, k2 = 12∑i

(3z2i − r2i )

1. The electric dipole: dipole moment:~p = Ql~e, where~e goes from⊕ to , and ~F = (~p · ∇) ~Eext, andW = −~p · ~Eout.

Electric field: ~E ≈ Q4πεr3

(3~p · ~rr2− ~p)

. The torque is:~τ = ~p× ~Eout

2. The magnetic dipole: dipole moment: ifr �√A: ~µ = ~I × (A~e⊥), ~F = (~µ · ∇) ~Bout

|µ| = mv2⊥

2B,W = −~µ× ~Bout

Magnetic field:~B =−µ4πr3

(3µ · ~rr2− ~µ

). The moment is:~τ = ~µ× ~Bout

2.7 Electric currents

The continuity equation for charge is:∂ρ

∂t+∇ · ~J = 0. Theelectric currentis given by:

I =dQ

dt=∫∫

( ~J · ~n )d2A

For most conductors holds:~J = ~E/ρ, whereρ is theresistivity.


If the flux enclosed by a conductor changes this results in aninduced voltageVind = −NdΦdt

. If the current

flowing through a conductor changes, this results in a self-inductance which opposes the original change:

Vselfind = −LdI

dt. If a conductor encloses a fluxΦ holds:Φ = LI.

The magnetic induction within a coil is approximated by:B =µNI√l2 + 4R2

wherel is the length,R the radius

andN the number of coils. The energy contained within a coil is given byW = 12LI2 andL = µN2A/l.

Thecapacityis defined by:C = Q/V . For a capacitor holds:C = ε0εrA/d whered is the distance betweenthe plates andA the surface of one plate. The electric field strength between the plates isE = σ/ε0 = Q/ε0Awhereσ is the surface charge. The accumulated energy is given byW = 12CV

2. The current through a

capacity is given byI = −C dVdt

.

For most PTC resistors holds approximately:R = R0(1 + αT ), whereR0 = ρl/A. For a NTC holds:R(T ) = C exp(−B/T ) whereB andC depend only on the material.

If a current flows through two different, connecting conductorsx andy, the contact area will heat up or cooldown, depending on the direction of the current: thePeltier effect. The generated or removed heat is given by:W = ΠxyIt. This effect can be amplified with semiconductors.

The thermic voltagebetween 2 metals is given by:V = γ(T − T0). For a Cu-Konstantane connection holds:γ ≈ 0.2− 0.7 mV/K.

In an electrical net with only stationary currents,Kirchhoff ’s equations apply: for a knot holds:∑In = 0,

along a closed path holds:∑Vn =

∑InRn = 0.

2.8 Depolarizing field

If a dielectric material is placed in an electric or magnetic field, the field strength within and outside thematerial will change because the material will be polarized or magnetized. If the medium has an ellipsoidalshape and one of the principal axes is parallel with the external field~E0 or ~B0 then the depolarizing is fieldhomogeneous.

~Edep = ~Emat − ~E0 = −N ~Pε0

~Hdep = ~Hmat − ~H0 = −N ~M

N is a constant depending only on the shape of the object placed in the field, with0 ≤ N ≤ 1. For a fewlimiting cases of an ellipsoid holds: a thin plane:N = 1, a long, thin bar:N = 0, a sphere:N = 13 .

2.9 Mixtures of materials

The average electric displacement in a material which is inhomogenious on a mesoscopic scale is given by:

〈D〉 = 〈εE〉 = ε∗ 〈E〉 whereε∗ = ε1(

1− φ2(1− x)Φ(ε∗/ε2)

)−1wherex = ε1/ε2. For a sphere holds:Φ =

13 +

23x. Further holds: (∑

i

φiεi

)−1≤ ε∗ ≤

∑i

φiεi

Chapter 3

Relativity

3.1 Special relativity

3.1.1 The Lorentz transformation

The Lorentz transformation(~x ′, t′) = (~x ′(~x, t), t′(~x, t)) leaves the wave equation invariant ifc is invariant:

∂2

∂x2+

∂2

∂y2+

∂2

∂z2− 1c2∂2

∂t2=

∂2

∂x′2+

∂2

∂y′2+

∂2

∂z′2− 1c2

∂2

∂t′2

This transformation can also be found whends2 = ds′2 is demanded. The general form of the Lorentztransformation is given by:

~x ′ = ~x+(γ − 1)(~x · ~v )~v

|v|2− γ~vt , t′ = γ

(t− ~x · ~v

c2

)where

γ =1√

1− v2

c2

The velocity difference~v ′ between two observers transforms according to:

~v ′ =(γ

(1− ~v1 · ~v2

c2

))−1(~v2 + (γ − 1)

~v1 · ~v2v21

~v1 − γ~v1)

If the velocity is parallel to thex-axis, this becomesy′ = y, z′ = z and:

x′ = γ(x− vt) , x = γ(x′ + vt′)

t′ = γ(t− xv

c2

), t = γ

(t′ +

x′v

c2

), v′ =

v2 − v11− v1v2

c2

If ~v = v~ex holds:

p′x = γ(px −

βW

c

), W ′ = γ(W − vpx)

With β = v/c the electric field of a moving charge is given by:

~E =Q

4πε0r2(1− β2)~er

(1− β2 sin2(θ))3/2

The electromagnetic field transforms according to:

~E′ = γ( ~E + ~v × ~B ) , ~B′ = γ

(~B − ~v ×

~E

c2

)

Length, mass and time transform according to:∆tr = γ∆t0, mr = γm0, lr = l0/γ, with 0 the quantitiesin a co-moving reference frame andr the quantities in a frame moving with velocityv w.r.t. it. The propertime τ is defined as:dτ2 = ds2/c2, so∆τ = ∆t/γ. For energy and momentum holds:W = mrc2 = γW0,

13


W 2 = m20c4 + p2c2. p = mrv = γm0v = Wv/c2, andpc = Wβ whereβ = v/c. The force is definedby

~F = d~p/dt.

4-vectors have the property that their modulus is independent of the observer: their componentscanchangeafter a coordinate transformation but not their modulus. The difference of two 4-vectors transforms also as

a 4-vector. The 4-vector for the velocity is given byUα =dxα

dτ. The relation with the “common” velocity

ui := dxi/dt is: Uα = (γui, icγ). For particles with nonzero restmass holds:UαUα = −c2, for particleswith zero restmass (so withv = c) holds:UαUα = 0. The 4-vector for energy and momentum is given by:pα = m0Uα = (γpi, iW/c). So:pαpα = −m20c2 = p2 −W 2/c2.

3.1.2 Red and blue shift

There are three causes of red and blue shifts:

1. Motion: with~ev · ~er = cos(ϕ) follows:f ′

f= γ

(1− v cos(ϕ)

c

).

This can give both red- and blueshift, also⊥ to the direction of motion.

2. Gravitational redshift:∆ff

=κM

rc2.

3. Redshift because the universe expands, resulting in e.g. the cosmic background radiation:λ0λ1

=R0R1

.

3.1.3 The stress-energy tensor and the field tensor

The stress-energy tensor is given by:

Tµν = (%c2 + p)uµuν + pgµν +1c2(FµαF

αν +

14gµνF

αβFαβ)

The conservation laws can than be written as:∇νTµν = 0. The electromagnetic field tensor is given by:

Fαβ =∂Aβ∂xα

− ∂Aα∂xβ

with Aµ := ( ~A, iV/c) andJµ := ( ~J, icρ). The Maxwell equations can than be written as:

∂νFµν = µ0Jµ , ∂λFµν + ∂µFνλ + ∂νFλµ = 0

The equations of motion for a charged particle in an EM field become with the field tensor:

dpαdτ

= qFαβuβ

3.2 General relativity

3.2.1 Riemannian geometry, the Einstein tensor

The basic principles of general relativity are:

1. The geodesic postulate: free falling particles move along geodesics of space-time with the proper timeτ or arc lengths as parameter. For particles with zero rest mass (photons), the use of a free parameter isrequired because for them holdsds = 0. Fromδ

∫ds = 0 the equations of motion can be derived:

d2xα

ds2+ Γαβγ

dxβ

ds

dxγ

ds= 0

Chapter 3: Relativity 15

2. Theprinciple of equivalence: inertial mass≡ gravitational mass⇒ gravitation is equivalent with acurved space-time were particles move along geodesics.

3. By a proper choice of the coordinate system it is possible to make the metric locally flat in each pointxi: gαβ(xi) = ηαβ :=diag(−1, 1, 1, 1).

TheRiemann tensoris defined as:RµναβTν := ∇α∇βTµ−∇β∇αTµ, where the covariant derivative is given

by∇jai = ∂jai + Γijkak and∇jai = ∂jai − Γkijak. Here,

Γijk =gil

2

(∂glj∂xk

+∂glk∂xj

−∂g

jk

∂xl

), for Euclidean spaces this reduces to:Γijk =

∂2x̄l

∂xj∂xk∂xi

∂x̄l,

are theChristoffel symbols. For a second-order tensor holds:[∇α,∇β ]Tµν = RµσαβT

σν + R

σναβT

µσ , ∇kaij =

∂kaij−Γlkjail +Γiklalj ,∇kaij = ∂kaij−Γlkialj−Γlkjajl and∇kaij = ∂kaij +Γiklalj +Γ

jkla

il. The followingholds:Rαβµν = ∂µΓ

αβν − ∂νΓαβµ + ΓασµΓσβν − ΓασνΓσβµ.

TheRicci tensoris a contraction of the Riemann tensor:Rαβ := Rµαµβ , which is symmetric:Rαβ = Rβα.

TheBianchi identitiesare:∇λRαβµν +∇νRαβλµ +∇µRαβνλ = 0.

The Einstein tensoris given by: Gαβ := Rαβ − 12gαβR, whereR := Rαα is theRicci scalar, for which

holds: ∇βGαβ = 0. With the variational principleδ∫

(L(gµν) − Rc2/16πκ)√|g|d4x = 0 for variations

gµν → gµν + δgµν theEinstein field equationscan be derived:

Gαβ =8πκc2

Tαβ , which can also be written asRαβ =8πκc2

(Tαβ − 12gαβTµµ )

For empty space this is equivalent toRαβ = 0. The equationRαβµν = 0 has as only solution a flat space.

The Einstein equations are 10 independent equations, which are of second order ingµν . From this, the Laplaceequation from Newtonian gravitation can be derived by stating:gµν = ηµν + hµν , where|h| � 1. In thestationary case, this results in∇2h00 = 8πκ%/c2.

The most general form of the field equations is:Rαβ − 12gαβR+ Λgαβ =8πκc2

Tαβ

whereΛ is thecosmological constant. This constant plays a role in inflatory models of the universe.

3.2.2 The line element

Themetric tensorin an Euclidean space is given by:gij =∑

k

∂x̄k

∂xi∂x̄k

∂xj.

In general holds:ds2 = gµνdxµdxν . In special relativity this becomesds2 = −c2dt2 + dx2 + dy2 + dz2.This metric,ηµν :=diag(−1, 1, 1, 1), is called theMinkowski metric.

Theexternal Schwarzschild metricapplies in vacuum outside a spherical mass distribution, and is given by:

ds2 =(−1 + 2m

r

)c2dt2 +

(1− 2m

r

)−1dr2 + r2dΩ2

Here,m := Mκ/c2 is thegeometrical massof an object with massM , anddΩ2 = dθ2 + sin2 θdϕ2. Thismetric is singular forr = 2m = 2κM/c2. If an object is smaller than its event horizon2m, that implies thatits escape velocity is> c, it is called ablack hole. The Newtonian limit of this metric is given by:

ds2 = −(1 + 2V )c2dt2 + (1− 2V )(dx2 + dy2 + dz2)

whereV = −κM/r is the Newtonian gravitation potential. In general relativity, the components ofgµν areassociated with the potentials and the derivatives ofgµν with the field strength.

The Kruskal-Szekeres coordinates are used to solve certain problems with the Schwarzschild metric nearr = 2m. They are defined by:


• r > 2m: u =

√r

2m− 1 exp

( r4m

)cosh

(t

4m

)

v =√

r

2m− 1 exp

( r4m

)sinh

(t

4m

)• r < 2m:

u =√

1− r2m

exp( r

4m

)sinh

(t

4m

)

v =√

1− r2m

exp( r

4m

)cosh

(t

4m

)• r = 2m: here, the Kruskal coordinates are singular, which is necessary to eliminate the coordinate

singularity there.

The line element in these coordinates is given by:

ds2 = −32m3

re−r/2m(dv2 − du2) + r2dΩ2

The liner = 2m corresponds tou = v = 0, the limit x0 →∞ with u = v andx0 → −∞ with u = −v. TheKruskal coordinates are only singular on the hyperbolev2 − u2 = 1, this corresponds withr = 0. On the linedv = ±du holdsdθ = dϕ = ds = 0.For the metric outside a rotating, charged spherical mass the Newman metric applies:

ds2 =(

1− 2mr − e2

r2 + a2 cos2 θ

)c2dt2 −

(r2 + a2 cos2 θ

r2 − 2mr + a2 − e2

)dr2 − (r2 + a2 cos2 θ)dθ2 −(

r2 + a2 +(2mr − e2)a2 sin2 θr2 + a2 cos2 θ

)sin2 θdϕ2 +

(2a(2mr − e2)r2 + a2 cos2 θ

)sin2 θ(dϕ)(cdt)

wherem = κM/c2, a = L/Mc ande = κQ/ε0c2.A rotating charged black hole has an event horizon withRS = m+

√m2 − a2 − e2.

Near rotating black holes frame dragging occurs becausegtϕ 6= 0. For the Kerr metric (e = 0, a 6= 0) thenfollows that within the surfaceRE = m+

√m2 − a2 cos2 θ (de ergosphere) no particle can be at rest.

3.2.3 Planetary orbits and the perihelion shift

To find a planetary orbit, the variational problemδ∫ds = 0 has to be solved. This is equivalent to the problem

δ∫ds2 = δ

∫gijdx

idxj = 0. Substituting the external Schwarzschild metric yields for a planetary orbit:

du

dϕ

(d2u

dϕ2+ u)

=du

dϕ

(3mu+

m

h2

)whereu := 1/r andh = r2ϕ̇ =constant. The term3mu is not present in the classical solution. This term can

in the classical case also be found from a potentialV (r) = −κMr

(1 +

h2

r2

).

The orbital equation givesr =constant as solution, or can, after dividing bydu/dϕ, be solved with perturbationtheory. In zeroth order, this results in an elliptical orbit:u0(ϕ) = A + B cos(ϕ) with A = m/h2 andB anarbitrary constant. In first order, this becomes:

u1(ϕ) = A+B cos(ϕ− εϕ) + ε(A+

B2

2A− B

2

6Acos(2ϕ)

)whereε = 3m2/h2 is small. The perihelion of a planet is the point for whichr is minimal, oru maximal.This is the case ifcos(ϕ− εϕ) = 0 ⇒ ϕ ≈ 2πn(1 + ε). For the perihelion shift then follows:∆ϕ = 2πε =6πm2/h2 per orbit.

Chapter 3: Relativity 17

3.2.4 The trajectory of a photon

For the trajectory of a photon (and for each particle with zero restmass) holdsds2 = 0. Substituting theexternal Schwarzschild metric results in the following orbital equation:

du

dϕ

(d2u

dϕ2+ u− 3mu

)= 0

3.2.5 Gravitational waves

Starting with the approximationgµν = ηµν + hµν for weak gravitational fields and the definitionh′µν =hµν − 12ηµνh

αα it follows that h

′µν = 0 if the gauge condition∂h

′µν/∂x

ν = 0 is satisfied. From this, itfollows that the loss of energy of a mechanical system, if the occurring velocities are� c and for wavelengths� the size of the system, is given by:

dE

dt= − G

5c5∑i,j

(d3Qijdt3

)2with Qij =

∫%(xixj − 13δijr

2)d3x the mass quadrupole moment.

3.2.6 Cosmology

If for the universe as a whole is assumed:

1. There exists a global time coordinate which acts asx0 of a Gaussian coordinate system,

2. The 3-dimensional spaces are isotrope for a certain value ofx0,

3. Each point is equivalent to each other point for a fixedx0.

then theRobertson-Walker metriccan be derived for the line element:

ds2 = −c2dt2 + R2(t)

r20

(1− kr

2

4r20

) (dr2 + r2dΩ2)For thescalefactorR(t) the following equations can be derived:

2R̈R

+Ṙ2 + kc2

R2= −8πκp

c2+ Λ and

Ṙ2 + kc2

R2=

8πκ%3

+Λ3

wherep is the pressure and% the density of the universe. IfΛ = 0 can be derived for thedecelerationparameterq:

q = − R̈RṘ2

=4πκ%3H2

whereH = Ṙ/R is Hubble’s constant. This is a measure of the velocity with which galaxies far away aremoving away from each other, and has the value≈ (75±25) km·s−1·Mpc−1. This gives 3 possible conditionsfor the universe (here,W is the total amount of energy in the universe):

1. Parabolical universe: k = 0, W = 0, q = 12 . The expansion velocity of the universe→ 0 if t → ∞.The hereto relatedcritical densityis %c = 3H2/8πκ.

2. Hyperbolical universe: k = −1, W < 0, q < 12 . The expansion velocity of the universe remainspositive forever.

3. Elliptical universe: k = 1, W > 0, q > 12 . The expansion velocity of the universe becomes negativeafter some time: the universe starts collapsing.

Chapter 4

Oscillations

4.1 Harmonic oscillations

The general form of a harmonic oscillation is:Ψ(t) = Ψ̂ei(ωt±ϕ) ≡ Ψ̂ cos(ωt± ϕ),

whereΨ̂ is theamplitude. A superposition of several harmonic oscillationswith the same frequencyresults inanother harmonic oscillation: ∑

i

Ψ̂i cos(αi ± ωt) = Φ̂ cos(β ± ωt)

with:

tan(β) =

∑i

Ψ̂i sin(αi)∑i

Ψ̂i cos(αi)and Φ̂2 =

∑i

Ψ̂2i + 2∑j>i

∑i

Ψ̂iΨ̂j cos(αi − αj)

For harmonic oscillations holds:∫x(t)dt =

x(t)iω

anddnx(t)dtn

= (iω)nx(t).

4.2 Mechanic oscillations

For a construction with a spring with constantC parallel to a dampingk which is connected to a massM , towhich a periodic forceF (t) = F̂ cos(ωt) is applied holds the equation of motionmẍ = F (t) − kẋ − Cx.With complex amplitudes, this becomes−mω2x = F − Cx− ikωx. With ω20 = C/m follows:

x =F

m(ω20 − ω2) + ikω,and for the velocity holds:̇x =

F

i√Cmδ + k

whereδ =ω

ω0− ω0ω

. The quantityZ = F/ẋ is called theimpedanceof the system. Thequalityof the system

is given byQ =√Cm

k.

The frequency with minimal|Z| is calledvelocity resonance frequency. This is equal toω0. In theresonancecurve|Z|/

√Cm is plotted againstω/ω0. The width of this curve is characterized by the points where|Z(ω)| =

|Z(ω0)|√

2. In these points holds:R = X andδ = ±Q−1, and the width is2∆ωB = ω0/Q.

Thestiffnessof an oscillating system is given byF/x. Theamplitude resonance frequencyωA is the frequency

whereiωZ is minimal. This is the case forωA = ω0√

1− 12Q2.

Thedamping frequencyωD is a measure for the time in which an oscillating system comes to rest. It is given

by ωD = ω0

√1− 1

4Q2. A weak damped oscillation(k2 < 4mC) dies out afterTD = 2π/ωD. For acritical

dampedoscillation(k2 = 4mC) holdsωD = 0. A strong damped oscillation(k2 > 4mC) drops like (ifk2 � 4mC) x(t) ≈ x0 exp(−t/τ).

4.3 Electric oscillations

The impedanceis given by: Z = R + iX. The phase angle isϕ := arctan(X/R). The impedance of aresistor isR, of a capacitor1/iωC and of a self inductoriωL. The quality of a coil isQ = ωL/R. The totalimpedance in case several elements are positioned is given by:

18

Chapter 4: Oscillations 19

1. Series connection:V = IZ,

Ztot =∑

i

Zi , Ltot =∑

i

Li ,1Ctot

=∑

i

1Ci

, Q =Z0R

, Z = R(1 + iQδ)

2. parallel connection:V = IZ,

1Ztot

=∑

i

1Zi

,1Ltot

=∑

i

1Li

, Ctot =∑

i

Ci , Q =R

Z0, Z =

R

1 + iQδ

Here,Z0 =

√L

Candω0 =

1√LC

.

The power given by a source is given byP (t) = V (t) · I(t), so〈P 〉t = V̂eff Îeff cos(∆φ)= 12 V̂ Î cos(φv − φi) =

12 Î

2Re(Z) = 12 V̂2Re(1/Z), wherecos(∆φ) is the work factor.

4.4 Waves in long conductors

These cables are in use for signal transfer, e.g. coax cable. For them holds:Z0 =

√dL

dx

dx

dC.

The transmission velocity is given byv =

√dx

dL

dx

dC.

4.5 Coupled conductors and transformers

For two coils enclosing each others flux holds: ifΦ12 is the part of the flux originating fromI2 through coil 2which is enclosed by coil 1, than holdsΦ12 = M12I2, Φ21 = M21I1. For the coefficients of mutual inductionMij holds:

M12 = M21 := M = k√L1L2 =

N1Φ1I2

=N2Φ2I1

∼ N1N2

where0 ≤ k ≤ 1 is thecoupling factor. For a transformer isk ≈ 1. At full load holds:

V1V2

=I2I1

= − iωMiωL2 +Rload

≈ −√L1L2

= −N1N2

4.6 Pendulums

The oscillation timeT = 1/f , and for different types of pendulums is given by:

• Oscillating spring:T = 2π√m/C if the spring force is given byF = C ·∆l.

• Physical pendulum:T = 2π√I/τ with τ the moment of force andI the moment of inertia.

• Torsion pendulum:T = 2π√I/κwith κ =

2lmπr4∆ϕ

the constant of torsion andI the moment of inertia.

• Mathematical pendulum:T = 2π√l/g with g the acceleration of gravity andl the length of the pendu-

lum.

Chapter 5

Waves

5.1 The wave equation

The general form of the wave equation is:u = 0, or:

∇2u− 1v2∂2u

∂t2=∂2u

∂x2+∂2u

∂y2+∂2u

∂z2− 1v2∂2u

∂t2= 0

whereu is the disturbance andv the propagation velocity. In general holds:v = fλ. By definition holds:kλ = 2π andω = 2πf .

In principle, there are two types of waves:

1. Longitudinal waves: for these holds~k ‖ ~v ‖ ~u.

2. Transversal waves: for these holds~k ‖ ~v ⊥ ~u.

Thephase velocityis given byvph = ω/k. Thegroup velocityis given by:

vg =dω

dk= vph + k

dvphdk

= vph

(1− k

n

dn

dk

)wheren is the refractive index of the medium. Ifvph does not depend onω holds:vph = vg. In a dispersivemedium it is possible thatvg > vph or vg < vph, andvg · vf = c2. If one wants to transfer information witha wave, e.g. by modulation of an EM wave, the information travels with the velocity at with a change in theelectromagnetic field propagates. This velocity is often almost equal to the group velocity.

For some media, the propagation velocity follows from:

• Pressure waves in a liquid or gas:v =√κ/%, whereκ is the modulus of compression.

• For pressure waves in a gas also holds:v =√γp/% =

√γRT/M .

• Pressure waves in a thin solid bar with diameter

Chapter 5: Waves 21

The equation for a harmonic traveling plane wave is:u(~x, t) = û cos(~k · ~x± ωt+ ϕ)If waves reflect at the end of a spring this will result in a change in phase. A fixed end gives a phase change ofπ/2 to the reflected wave, with boundary conditionu(l) = 0. A lose end gives no change in the phase of thereflected wave, with boundary condition(∂u/∂x)l = 0.

If an observer is moving w.r.t. the wave with a velocityvobs, he will observe a change in frequency: the

Doppler effect. This is given by:f

f0=vf − vobs

vf.

5.2.2 Spherical waves

When the situation is spherical symmetric, the homogeneous wave equation is given by:

1v2∂2(ru)∂t2

− ∂2(ru)∂r2

= 0

with general solution:

u(r, t) = C1f(r − vt)

r+ C2

g(r + vt)r

5.2.3 Cylindrical waves

When the situation has a cylindrical symmetry, the homogeneous wave equation becomes:

1v2∂2u

∂t2− 1r

∂

∂r

(r∂u

∂r

)= 0

This is a Bessel equation, with solutions which can be written as Hankel functions. For sufficient large valuesof r these are approximated by:

u(r, t) =û√r

cos(k(r ± vt))

5.2.4 The general solution in one dimension

Starting point is the equation:

∂2u(x, t)∂t2

=N∑

m=0

(bm

∂m

∂xm

)u(x, t)

wherebm ∈ IR. Substitutingu(x, t) = Aei(kx−ωt) gives two solutionsωj = ωj(k) as dispersion relations.The general solution is given by:

u(x, t) =

∞∫−∞

(a(k)ei(kx−ω1(k)t) + b(k)ei(kx−ω2(k)t)

)dk

Because in general the frequenciesωj are non-linear ink there is dispersion and the solution cannot be writtenany more as a sum of functions depending only onx± vt: the wave front transforms.

5.3 The stationary phase method

Usually the Fourier integrals of the previous section cannot be calculated exactly. Ifωj(k) ∈ IR the stationaryphase method can be applied. Assuming thata(k) is only a slowly varying function ofk, one can state that theparts of thek-axis where the phase ofkx− ω(k)t changes rapidly will give no net contribution to the integralbecause the exponent oscillates rapidly there. The only areas contributing significantly to the integral are areas

with a stationary phase, determined byd

dk(kx− ω(k)t) = 0. Now the following approximation is possible:

∞∫−∞

a(k)ei(kx−ω(k)t)dk ≈N∑

i=1

√√√√ 2πd2ω(ki)

dk2i

exp[−i14π + i(kix− ω(ki)t)

]


5.4 Green functions for the initial-value problem

This method is preferable if the solutions deviate much from the stationary solutions, like point-like excitations.Starting with the wave equation in one dimension, with∇2 = ∂2/∂x2 holds: ifQ(x, x′, t) is the solution with

initial valuesQ(x, x′, 0) = δ(x − x′) and ∂Q(x, x′, 0)

∂t= 0, andP (x, x′, t) the solution with initial values

P (x, x′, 0) = 0 and∂P (x, x′, 0)

∂t= δ(x − x′), then the solution of the wave equation with arbitrary initial

conditionsf(x) = u(x, 0) andg(x) =∂u(x, 0)∂t

is given by:

u(x, t) =

∞∫−∞

f(x′)Q(x, x′, t)dx′ +

∞∫−∞

g(x′)P (x, x′, t)dx′

P andQ are called thepropagators. They are defined by:

Q(x, x′, t) = 12 [δ(x− x′ − vt) + δ(x− x′ + vt)]

P (x, x′, t) =

{ 12v

if |x− x′| < vt0 if |x− x′| > vt

Further holds the relation:Q(x, x′, t) =∂P (x, x′, t)

∂t

5.5 Waveguides and resonating cavities

The boundary conditions for a perfect conductor can be derived from the Maxwell equations. If~n is a unitvector⊥ the surface, pointed from 1 to 2, and~K is a surface current density, than holds:

~n · ( ~D2 − ~D1) = σ ~n× ( ~E2 − ~E1) = 0~n · ( ~B2 − ~B1) = 0 ~n× ( ~H2 − ~H1) = ~K

In a waveguide holds because of the cylindrical symmetry:~E(~x, t) = ~E(x, y)ei(kz−ωt) and ~B(~x, t) =~B(x, y)ei(kz−ωt). From this one can now deduce that, ifBz andEz are not≡ 0:

Bx =i

εµω2 − k2

(k∂Bz∂x− εµω∂Ez

∂y

)By =

i

εµω2 − k2

(k∂Bz∂y

+ εµω∂Ez∂x

)Ex =

i

εµω2 − k2

(k∂Ez∂x

+ εµω∂Bz∂y

)Ey =

i

εµω2 − k2

(k∂Ez∂y− εµω∂Bz

∂x

)Now one can distinguish between three cases:

1. Bz ≡ 0: the Transversal Magnetic modes (TM). Boundary condition:Ez|surf = 0.

2. Ez ≡ 0: the Transversal Electric modes (TE). Boundary condition:∂Bz∂n

∣∣∣∣surf

= 0.

For the TE and TM modes this gives an eigenvalue problem forEz resp.Bz with boundary conditions:(∂2

∂x2+

∂2

∂y2

)ψ = −γ2ψ with eigenvaluesγ2 := εµω2 − k2

This gives a discrete solutionψ` with eigenvalueγ2` : k =√εµω2 − γ2` . Forω < ω`, k is imaginary

and the wave is damped. Therefore,ω` is called thecut-off frequency. In rectangular conductors thefollowing expression can be found for the cut-off frequency for modes TEm,n of TMm,n:

λ` =2√

(m/a)2 + (n/b)2

Chapter 5: Waves 23

3. Ez andBz are zero everywhere: the Transversal electromagnetic mode (TEM). Than holds:k =±ω√εµ andvf = vg, just as if here were no waveguide. Furtherk ∈ IR, so there exists no cut-offfrequency.

In a rectangular, 3 dimensional resonating cavity with edgesa, b andc the possible wave numbers are given

by: kx =n1π

a, ky =

n2π

b, kz =

n3π

cThis results in the possible frequenciesf = vk/2π in the cavity:

f =v

2

√n2xa2

+n2yb2

+n2zc2

For a cubic cavity, witha = b = c, the possible number of oscillating modesNL for longitudinal waves isgiven by:

NL =4πa3f3

3v3

Because transversal waves have two possible polarizations holds for them:NT = 2NL.

5.6 Non-linear wave equations

TheVan der Polequation is given by:

d2x

dt2− εω0(1− βx2)

dx

dt+ ω20x = 0

βx2 can be ignored for very small values of the amplitude. Substitution ofx ∼ eiωt gives: ω = 12ω0(iε ±2√

1− 12ε2). The lowest-order instabilities grow as12εω0. While x is growing, the 2nd term becomes larger

and diminishes the growth. Oscillations on a time scale∼ ω−10 can exist. Ifx is expanded asx = x(0) +εx(1) + ε2x(2) + · · · and this is substituted one obtains, besides periodic,secular terms∼ εt. If it is assumedthat there exist timescalesτn, 0 ≤ τ ≤ N with ∂τn/∂t = εn and if the secular terms are put 0 one obtains:

d

dt

{12

(dx

dt

)2+ 12ω

20x

2

}= εω0(1− βx2)

(dx

dt

)2This is an energy equation. Energy is conserved if the left-hand side is 0. Ifx2 > 1/β, the right-hand sidechanges sign and an increase in energy changes into a decrease of energy. This mechanism limits the growthof oscillations.

TheKorteweg-De Vriesequation is given by:

∂u

∂t+∂u

∂x− au∂u

∂x︸︷︷︸non−lin

+ b2∂3u

∂x3︸︷︷︸dispersive

= 0

This equation is for example a model for ion-acoustic waves in a plasma. For this equation, soliton solutionsof the following form exist:

u(x− ct) = −dcosh2(e(x− ct))

with c = 1 + 13ad ande2 = ad/(12b2).

Chapter 6

Optics

6.1 The bending of light

For the refraction at a surface holds:ni sin(θi) = nt sin(θt) wheren is therefractive indexof the material.Snell’s law is:

n2n1

=λ1λ2

=v1v2

If ∆n ≤ 1, the change in phase of the light is∆ϕ = 0, if ∆n > 1 holds:∆ϕ = π. The refraction of light in amaterial is caused by scattering from atoms. This is described by:

n2 = 1 +nee

2

ε0m

∑j

fjω20,j − ω2 − iδω

wherene is the electron density andfj theoscillator strength, for which holds:∑j

fj = 1. From this follows

thatvg = c/(1 + (nee2/2ε0mω2)). From this the equation of Cauchy can be derived:n = a0 + a1/λ2. More

general, it is possible to expandn as:n =n∑

k=0

akλ2k

.

For an electromagnetic wave in general holds:n =√εrµr.

The path, followed by a light ray in material can be found fromFermat’s principle:

δ

2∫1

dt = δ

2∫1

n(s)cds = 0⇒ δ

2∫1

n(s)ds = 0

6.2 Paraxial geometrical optics

6.2.1 Lenses

The Gaussian lens formula can be deduced from Fermat’s principle with the approximationscosϕ = 1 andsinϕ = ϕ. For the refraction at a spherical surface with radiusR holds:

n1v− n2

b=n1 − n2R

where|v| is the distance of the object and|b| the distance of the image. Applying this twice results in:

1f

= (nl − 1)(

1R2− 1R1

)wherenl is the refractive index of the lens,f is the focal length andR1 andR2 are the curvature radii of bothsurfaces. For a double concave lens holdsR1 < 0, R2 > 0, for a double convex lens holdsR1 > 0 andR2 < 0. Further holds:

1f

=1v− 1b

24

Chapter 6: Optics 25

D := 1/f is called the dioptric power of a lens. For a lens with thicknessd and diameterD holds to a goodapproximation:1/f = 8(n− 1)d/D2. For two lenses placed on a line with distanced holds:

1f

=1f1

+1f2− df1f2

In these equations the following signs are being used for refraction at a spherical surface, as is seen by anincoming light ray:

Quantity + −R Concave surface Convex surfacef Converging lens Diverging lensv Real object Virtual objectb Virtual image Real image

6.2.2 Mirrors

For images of mirrors holds:

1f

=1v

+1b

=2R

+h2

2

(1R− 1v

)2whereh is the perpendicular distance from the point the light ray hits the mirror to the optical axis. Sphericalaberration can be reduced by not using spherical mirrors. A parabolical mirror has no spherical aberration forlight rays parallel with the optical axis and is therefore often used for telescopes. The used signs are:

Quantity + −R Concave mirror Convex mirrorf Concave mirror Convex mirrorv Real object Virtual objectb Real image Virtual image

6.2.3 Principal planes

Thenodal pointsN of a lens are defined by the figure on the right. If the lens issurrounded by the same medium on both sides, the nodal points are the same asthe principal points H. The plane⊥ the optical axis through the principal pointsis called theprincipal plane. If the lens is described by a matrixmij than for thedistancesh1 andh2 to the boundary of the lens holds:

h1 = nm11 − 1m12

, h2 = nm22 − 1m12

r��,

,,

,��

rrN1N2O

6.2.4 Magnification

The linear magnificationis defined by:N = − bv

Theangular magnificationis defined by:Nα = −αsystαnone

whereαsys is the size of the retinal image with the optical system andαnone the size of the retinal imagewithout the system. Further holds:N ·Nα = 1. For a telescope holds:N = fobjective/focular. Thef-numberis defined byf/Dobjective.


6.3 Matrix methods

A light ray can be described by a vector(nα, y) with α the angle with the optical axis andy the distance tothe optical axis. The change of a light ray interacting with an optical system can be obtained using a matrixmultiplication: (

n2α2y2

)= M

(n1α1y1

)whereTr(M) = 1. M is a product of elementary matrices. These are:

1. Transfer along lengthl: MR =(

1 0l/n 1

)

2. Refraction at a surface with dioptric powerD: MT =(

1 −D0 1

)

6.4 Aberrations

Lenses usually do not give a perfect image. Some causes are:

1. Chromatic aberration is caused by the fact thatn = n(λ). This can be partially corrected with a lenswhich is composed of more lenses with different functionsni(λ). UsingN lenses makes it possible toobtain the samef for N wavelengths.

2. Spherical aberration is caused by second-order effects which are usually ignored; a spherical surfacedoes not make a perfect lens. Incomming rays far from the optical axis will more bent.

3. Coma is caused by the fact that the principal planes of a lens are only flat near the principal axis. Furtheraway of the optical axis they are curved. This curvature can be both positive or negative.

4. Astigmatism: from each point of an object not on the optical axis the image is an ellipse because thethickness of the lens is not the same everywhere.

5. Field curvature can be corrected by the human eye.

6. Distorsion gives abberations near the edges of the image. This can be corrected with a combination ofpositive and negative lenses.

6.5 Reflection and transmission

If an electromagnetic wave hits a transparent medium part of the wave will reflect at the same angle as theincident angle, and a part will be refracted at an angle according to Snell’s law. It makes a difference whetherthe ~E field of the wave is⊥ or ‖ w.r.t. the surface. When the coefficients of reflectionr and transmissiont aredefined as:

r‖ ≡(E0rE0i

)‖, r⊥ ≡

(E0rE0i

)⊥, t‖ ≡

(E0tE0i

)‖, t⊥ ≡

(E0tE0i

)⊥

whereE0r is the reflected amplitude andE0t the transmitted amplitude. Then the Fresnel equations are:

r‖ =tan(θi − θt)tan(θi + θt)

, r⊥ =sin(θt − θi)sin(θt + θi)

t‖ =2 sin(θt) cos(θi)

sin(θt + θi) cos(θt − θi), t⊥ =

2 sin(θt) cos(θi)sin(θt + θi)

The following holds:t⊥ − r⊥ = 1 andt‖ + r‖ = 1. If the coefficient of reflectionR and transmissionT aredefined as (withθi = θr):

R ≡ IrIi

and T ≡ It cos(θt)Ii cos(θi)

Chapter 6: Optics 27

with I = 〈|~S|〉 it follows: R+T = 1. A special case isr‖ = 0. This happens if the angle between the reflectedand transmitted rays is90◦. From Snell’s law it then follows:tan(θi) = n. This angle is calledBrewster’sangle. The situation withr⊥ = 0 is not possible.

6.6 Polarization

The polarization is defined as:P =Ip

Ip + Iu=Imax − IminImax + Imin

where the intensity of the polarized light is given byIp and the intensity of the unpolarized light is given byIu. Imax andImin are the maximum and minimum intensities when the light passes a polarizer. If polarizedlight passes through a polarizerMalus lawapplies:I(θ) = I(0) cos2(θ) whereθ is the angle of the polarizer.

The state of a light ray can be described by theStokes-parameters: start with 4 filters which each transmits halfthe intensity. The first is independent of the polarization, the second and third are linear polarizers with thetransmission axes horizontal and at+45◦, while the fourth is a circular polarizer which is opaque forL-states.Then holdsS1 = 2I1, S2 = 2I2 − 2I1, S3 = 2I3 − 2I1 andS4 = 2I4 − 2I1.The state of apolarizedlight ray can also be described by theJones vector:

~E =(E0xeiϕxE0yeiϕy

)For the horizontalP -state holds:~E = (1, 0), for the verticalP -state ~E = (0, 1), theR-state is given by~E = 12

√2(1,−i) and theL-state by~E = 12

√2(1, i). The change in state of a light beam after passage of

optical equipment can be described as~E2 = M · ~E1. For some types of optical equipment the Jones matrixMis given by:

Horizontal linear polarizer:

(1 00 0

)Vertical linear polarizer:

(0 00 1

)Linear polarizer at+45◦ 12

(1 11 1

)Lineair polarizer at−45◦ 12

(1 −1−1 1

)14 -λ plate, fast axis vertical e

iπ/4

(1 00 −i

)14 -λ plate, fast axis horizontal e

iπ/4

(1 00 i

)Homogene circular polarizor right 12

(1 i−i 1

)Homogene circular polarizer left 12

(1 −ii 1

)

6.7 Prisms and dispersion

A light ray passing through a prism is refracted twice and aquires a deviation from its original directionδ = θi + θi′ + α w.r.t. the incident direction, whereα is the apex angle,θi is the angle between the incidentangle and a line perpendicular to the surface andθi′ is the angle between the ray leaving the prism and a lineperpendicular to the surface. Whenθi varies there is an angle for whichδ becomes minimal. For the refractiveindex of the prism now holds:

n =sin( 12 (δmin + α))

sin( 12α)


The dispersion of a prism is defined by:

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Physics FormularyPhysics Formulary by ir. J.C.A. Wevers III 7.6 Interaction between molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 8 Thermodynamics